View source for Totally-irreducible set

A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t0934701.png" /> of linear operators (cf. [[Linear operator|Linear operator]]) on a locally convex [[Topological vector space|topological vector space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t0934702.png" /> that is everywhere dense in the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t0934703.png" /> of all weakly-continuous linear operators on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t0934704.png" /> (cf. [[Weak topology|Weak topology]]); in this context <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t0934705.png" /> is considered with the weak operator topology. The concept of a totally-irreducible set, initially introduced for Banach spaces, also proved useful in the theory of group representations, mainly for semi-simple Lie groups. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t0934706.png" /> is a totally-irreducible set, it is also topologically irreducible, i.e. any closed subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t0934707.png" /> which is invariant with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t0934708.png" /> coincides with zero or with the entire space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t0934709.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347010.png" /> is a totally-irreducible set, its commutator subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347011.png" /> consists of the operators that are multiples of the identity. The property of total irreducibility is equivalent to that of topological irreducibility in the following cases: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347012.png" />; or 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347013.png" /> is a semi-group of unitary operators on a Hilbert space.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  D.P. Zhelobenko,   "Harmonic analysis of functions on semi-simple complex Lie groups" , Moscow  (1974)  (In Russian)</TD></TR></table>



====Comments====
A set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347014.png" /> of operators on a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347015.png" /> is algebraically irreducible if there are no proper subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347016.png" /> (i.e. except <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347017.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347018.png" />) that are invariant under all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347019.png" />. Cf. also [[Irreducible representation|Irreducible representation]]. A totally-irreducible set is also called completely irreducible.

The link between topological irreducibility and algebraic irreducibility and the statement that the commutator subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347020.png" /> consists of multiples of the identity is provided by the [[Schur lemma|Schur lemma]]. For a finite-dimensional space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347021.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347022.png" />, the algebraic (or topological) irreducibility of a set of operators implies that the commutator subset in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347023.png" /> is a division algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347024.png" /> in general, and equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347025.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347026.png" /> is algebraically closed. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347027.png" /> is not algebraically closed, this need not be the case. E.g., if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347028.png" />, the commutant of an irreducible representation of a group can be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347030.png" /> or the four-dimensional algebra of quaternions over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347031.png" />. Thus, the statement above about the equivalence for finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347032.png" /> between algebraic irreducibility (or, what is the same in this case, topological irreducibility) holds for algebraically closed ground fields but it does not hold for a non-algebraically closed ground field. For instance, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347033.png" /> is the collection of operators

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347034.png" /></td> </tr></table>

(the left regular representation of the quaternion algebra on itself), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347035.png" /> is algebraically irreducible, but its commutant algebra is the four-dimensional algebra of all matrices of the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347036.png" /></td> </tr></table>

(another copy inside <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347037.png" /> of the quaternion algebra, corresponding to the right regular representation).

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  O. Bratteli,   D.W. Robinson,   "Operator algebras and quantum statistical mechanics" , '''1''' , Springer  (1979)  pp. 47</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Dixmier,   "Les <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t093/t093470/t09347038.png" />-algèbres et leur représentations" , Gauthier-Villars  (1964)  pp. §2.3</TD></TR></table>