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Jordan-Lie-Banach algebra

Context

Algebra

Quantum physics

physics, mathematical physics, philosophy of physics

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theory (physics), model (physics)

experiment, measurement, computable physics

Contents

Idea

A Jordan–Lie–Banach algebra (or JLBJLB-algebra for short) is a topological algebra that behaves like a Poisson algebra, only that the commutative product is not required to form an associative algebra, but just a Jordan algebra. Hence a JLBJLB-algebra is a nonassociative Poisson algebra with topology.

JLB-algebras are the outcome of quantization of Poisson algebras. In traditional strict deformation quantization that outcome is regarded to be a non-commutative but associative C-star-algebra. But any such induces a JLB-algebra by letting the Jordan product be the symmetrized product and the Lie product the commutator (times i/2\mathrm{i}/2). There is a condition relating the associator of the JLB-algebra to the Lie bracket, that characterizes those JLB-algebras that come from non-commutative associative algebras, and in the usual definition of JLB-algebra this condition is required. In that case JLB-algebras are effectively the same as C *C^*-algebras, the only difference being that the single assocative product is explcitly regarded as inducing the two products of a non-associative Poisson algebra. For more on this separation of the Lie-algebra and the Jordan algebra aspect of quantization see at order-theoretic structure in quantum mechanics.

Definition

A JLB-algebra (over the real numbers) consists of a Banach space AA equipped with two short bilinear operators ()()(-)\circ(-) and ()()(-)\bullet(-), respectively called the Jordan product and the Lie product, satisfying the following identities:

  • Jordan commutativity: xy=yxx \circ y = y \circ x;
  • Lie anticommutativity: xx=0x \bullet x = 0, or equivalently (given bilinearity) xy=yxx \bullet y = -y \bullet x;
  • the Jacobi identity (Lie self-derivation): x(yz)=(xy)z+y(xz)x \bullet (y \bullet z) = (x \bullet y) \bullet z + y \bullet (x \bullet z), or equivalently (given anticommutativity) x(yz)+y(zx)+z(xy)=0x \bullet (y \bullet z) + y \bullet (z \bullet x) + z \bullet (x \bullet y) = 0;
  • Jordan derivation: x(yz)=(xy)z+y(xz)x \bullet (y \circ z) = (x \bullet y) \circ z + y \circ (x \bullet z);
  • the associator identity: x(yz)=(xy)z+y(xz)x \circ (y \circ z) = (x \circ y) \circ z + y \bullet (x \bullet z), or equivalently (given anticommutativity) (xy)zx(yz)=(xz)y(x \circ y) \circ z - x \circ (y \circ z) = (x \bullet z) \bullet y;
  • the BB-identity: xx=x 2{\|x \circ x\|} = {\|x\|^2} (compare the B *B^*-identity or C *C^*-identity of a C *C^*-algebra);
  • positivity: xxxx+yy{\|x \circ x \|} \leq {\|x \circ x + y \circ y\|}.

This definition is adapted from Section 1.1 of Halvorson, 1999. Halvorson does not include the statement that the Lie multiplication is short, and it includes a nonnegative real constant factor rr on the right-hand side of the associator identity (second version). However, Halvorson claims to construct an equivalence between real JLBJLB-algebras and complex C *C^*-algebras, and this construction produces a short Lie product that satisfies r=1r = 1.

Another consequence of this definition is that the Jordan product makes AA into a Jordan algebra (and hence into a JB-algebra).

Equivalence to C *C^*-algebras

The Jordan product and Lie product are respectively the real-symmetrized and imaginary-antisymmetrized parts of an associative operation on the complexification of AA, defining a complex C *C^*-algebra; and every C *C^*-algebra likewise defines a JLB-algebra consisting of its Hermitian elements.

Specifically, starting with a JLB-algebra AA, we write AAA \oplus A formally as A+iAA + \mathrm{i} A, on which we define the following operations:

  • norm: a+iba 2+b 2{\|a + \mathrm{i} b\|} \coloneqq \sqrt{{\|a\|^2} + {\|b\|^2}},
  • addition: (a+ib)+(c+id)(a+c)+i(b+d)(a + \mathrm{i} b) + (c + \mathrm{i} d) \coloneqq (a + c) + \mathrm{i} (b + d),
  • opposite: (a+ib)(a)+i(b)-(a + \mathrm{i} b) \coloneqq (-a) + \mathrm{i} (-b),
  • zero: 00+i00 \coloneqq 0 + \mathrm{i} 0,
  • scalar multiplication: (x+iy)(a+ib)=(xayb)+i(xb+ya)(x + \mathrm{i} y) (a + \mathrm{i} b) = (x a - y b) + \mathrm{i} (x b + y a) (for x+iyx + \mathrm{i} y a complex number),
  • involution: (a+ib) *a+i(b)(a + \mathrm{i} b)^* \coloneqq a + \mathrm{i} (-b),
  • multiplication: (a+ib)(c+id)(ac+ad+bcbd)+i(ac+ad+bc+bd)(a + \mathrm{i} b) (c + \mathrm{i} d) \coloneqq (a \circ c + a \bullet d + b \bullet c - b \circ d) + \mathrm{i} (-a \bullet c + a \circ d + b \circ c + b \bullet d).

If the Jordan product of the JLB-algebra has an identity 11, then so does the C *C^*-algebra:

  • 11+i01 \coloneqq 1 + \mathrm{i} 0.

Conversely, starting with a C *C^*-algebra AA, we form the subspace sa(A)={x:Ax *=x}sa(A) = \{x\colon A \;|\; x^* = x\}, on which we define the following operations (under each of which sa(A)sa(A) is closed):

  • norm, addition, opposite, zero, scalar multiplication (by real numbers only): by restriction,
  • Jordan product: ab12ab+12baa \circ b \coloneqq \frac{1}{2} a b + \frac{1}{2} b a,
  • Lie product: ab12iab12ibaa \bullet b \coloneqq \frac{1}{2} \mathrm{i} a b - \frac{1}{2} \mathrm{i} b a.

If the C *C^*-algebra has an identity, then this is also an identity for the Jordan product (so 11 is also defined by restriction).

This all defines a functor each way between the groupoids of C *C^*-algebras and JLBJLB-algebras, which in fact (I hope!) form an adjoint equivalence. Since we have a notion of morphism (not just isomorphism) of C *C^*-algebras, we can transport this along the equivalence to get a notion of morphism of JLBJLB-algebras (which I would expect to be a short linear map that preserves both products) and thus a category JLBAlgJLB Alg equivalent to C *AlgC^* Alg.

References

A definition is in section 1.1 of

A brief remark is on p. 80 of

Revised on October 14, 2013 09:33:30 by Toby Bartels (64.89.53.136)