Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia


Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 1 April 2010


An algebra is a vector space A with an associative multiplication A×A which satisfies the distributive laws, ie, such that A is a ring. A Banach algebra is a Banach space A with a multiplication such that A is an algebra and a 1 a 2 a 1 a 2 ,for all a 1 , a 2 A.

A * -algebra is a Banach algebra with an involution *:AA such that

An element a in a * -algebra is hermitian, or self-adjoint, if a*=a. A C* -algebra is a * -algebra A such that a*a- a 2 ,for allaA.

An idempotented algebra is an algebra A with a set of idempotents such that

  1. For each pair e 1 , e 2 there is an e 0 such that e 0 e 1 = e 1 e 0 = e 1 and e 0 e 2 = e 2 e 0 = e 2 , and
  2. For each aA there is an e such that ae=ea=a. A von Neumann algebra is an algebra A of operators on a Hilbert space H such that
      1. A is closed under taking adjoints,
      2. A coincides with its bicommutant.


  1. The algebra B H of bounded linear operators on a Hilbert space H with the operator norm ???? and involution given by adjoint ??? is a Banach algebra.
  2. Let G be a locally compact Hausdorff topological group G and let μ be a Haar measure on G. The vector space L 2 Gμ = f:G| f 2 < is a Hilbert space under the operations defined ????.
  3. Let V be a vector space. Then End V is an algebra.

References [PLACEHOLDER]

[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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